The use of the INLA approach is motivated by its computational benefits. The INLA method is a deterministic approach for doing Bayesian inference, as opposed to Markov Chaine Monte Carlo (MCMC) method which is a simulation-based approach [1]}. Rue et al. (2019) show that in particular spatial applications for which the size of the latent Gaussian field is large, the INLA method was able to compute accurate approximations of the posterior marginals which takes longer for MCMC algorithms to compute. In addition, the approximation bias is less than the MCMC error for typical examples. The SPDE approach also provides a means to speed up computation. For models using Gaussian random fields (GRF), which includes spatial and spatio-temporal models, there is a difficulty in the computation because of the need to factorize and invert typically large covariance matrices. The SPDE approach provides a Markov structure on the GRF. The Markov property causes the inverse covariance matrix, called the precision matrix, to be sparse. This sparsity makes the factorization of the precision matrix fast. Such models are called Gaussian Markov Random Fields (GMRF). In addition to making the computation efficient with the use of sparse matrix numerical algorithms because of the Markov structure, another benefit of using the SPDE approach is that it allows the interpretation of the model in terms of the parameters of the covariance function [2]}.
